Definition df-dmd 10164

Description: Define the dual modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M* for "the ordered pair <x,y> is a dual modular pair." See dmdbrt 10182 for membership relation.

Assertion

Ref	Expression
df-dmd	⊢ Mℋ* = {⟨x, y⟩∣((x ∈ Cℋ ⋀ y ∈ Cℋ ) ⋀ ∀z ∈ Cℋ (y ⊆ z → ((z ∩ x) ∨ℋ y) = (z ∩ (x ∨ℋ y))))}

Distinct variable group:   x,y,z

Detailed syntax breakdown of Definition df-dmd

Step	Hyp	Ref	Expression
1		cdmd 8791	. 2 class Mℋ*
2		vx	. . . . . . 7 set x
3	2	cv 954	. . . . . 6 class x
4		cch 8753	. . . . . 6 class Cℋ
5	3, 4	wcel 957	. . . . 5 wff x ∈ Cℋ
6		vy	. . . . . . 7 set y
7	6	cv 954	. . . . . 6 class y
8	7, 4	wcel 957	. . . . 5 wff y ∈ Cℋ
9	5, 8	wa 223	. . . 4 wff (x ∈ Cℋ ⋀ y ∈ Cℋ )
10		vz	. . . . . . . 8 set z
11	10	cv 954	. . . . . . 7 class z
12	7, 11	wss 2044	. . . . . 6 wff y ⊆ z
13	11, 3	cin 2043	. . . . . . . 8 class (z ∩ x)
14		chj 8757	. . . . . . . 8 class ∨ℋ
15	13, 7, 14	co 3958	. . . . . . 7 class ((z ∩ x) ∨ℋ y)
16	3, 7, 14	co 3958	. . . . . . . 8 class (x ∨ℋ y)
17	11, 16	cin 2043	. . . . . . 7 class (z ∩ (x ∨ℋ y))
18	15, 17	wceq 955	. . . . . 6 wff ((z ∩ x) ∨ℋ y) = (z ∩ (x ∨ℋ y))
19	12, 18	wi 3	. . . . 5 wff (y ⊆ z → ((z ∩ x) ∨ℋ y) = (z ∩ (x ∨ℋ y)))
20	19, 10, 4	wral 1643	. . . 4 wff ∀z ∈ Cℋ (y ⊆ z → ((z ∩ x) ∨ℋ y) = (z ∩ (x ∨ℋ y)))
21	9, 20	wa 223	. . 3 wff ((x ∈ Cℋ ⋀ y ∈ Cℋ ) ⋀ ∀z ∈ Cℋ (y ⊆ z → ((z ∩ x) ∨ℋ y) = (z ∩ (x ∨ℋ y))))
22	21, 2, 6	copab 2662	. 2 class {⟨x, y⟩∣((x ∈ Cℋ ⋀ y ∈ Cℋ ) ⋀ ∀z ∈ Cℋ (y ⊆ z → ((z ∩ x) ∨ℋ y) = (z ∩ (x ∨ℋ y))))}
23	1, 22	wceq 955	1 wff Mℋ* = {⟨x, y⟩∣((x ∈ Cℋ ⋀ y ∈ Cℋ ) ⋀ ∀z ∈ Cℋ (y ⊆ z → ((z ∩ x) ∨ℋ y) = (z ∩ (x ∨ℋ y))))}

This definition is referenced by:  dmdbrt 10182

Copyright terms: Public domain